5. Möbius Transformations

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "5. Möbius Transformations"

Transcription

1 5. Möbius Transformations 5.1. The linear transformation and the inversion. In this section we investigate the Möbius transformation which provides very convenient methods of finding a one-to-one mapping of one domain into another. Let us start with the a linear transformation where A and B are fixed complex numbers, A 0. We write (1) as w = φ(z) := Az + B, (1) w = φ(z) := A e iå(a) z + B. As we see this transformation is a composition of a rotation about the origin through the angle Arg (a) a magnification and a translation w 1 := e iarg (a) z, w 2 = A w 1 w = w 3 = w 2 + B. Each of these transformations are one-to-one mappings of the complex plane onto itself and gap geometric objects onto congruent objects. In particular, the the ranges of lines and of circles are line and circles, respectively. Now we consider the inversion defined by w := 1 z. (2) It is easy to see that the inversion is a one-to-one mapping of the extended complex plane C onto itself (0 and vice versa 0.) We are going to show that the image of a line is either a line or a circle. Indeed, let first l passes through the origin. The image of a point ρe iθ is 1 ρ e iθ. Letting ρ to tend from the negative infinity to the positive one, we see that the image is another line through the origin with an angle of inclination Θ. Let now L be given by the equation L : Ax + By = C, with C 0, and A + B > 0. (3) 1

2 On writing w = u + iv, we find and so x = z = w w = u iv 2 u 2 + v 2 u u 2 + v 2, y = Making these substitutions into (3), we get u u 2 + v 2. (4) or, equivalently, A u u 2 + v + B v 2 u 2 + v = C, 2 u 2 + v 2 A C u + B v = 0. (5) C This is apparently a equation of a circle. 5.2 We are prepared to go define a Möbius transformation in the general sense. Definition: The transformation w := az + b, a + c > 0, ad bc 0 cz + d is called a Möbius transformation. ℵ If c = 0, then the Möbius transformation is linear. If c 0, a = 0 then the transformation is a inversion. Consider the case ac 0. Then w can be written as w(z) = a ( ) bc ad 1 + (6) c a(cz + d) which is as a matter of fact a decomposition of a linear transformation and and inverse function. We also notice that w (z) = ad bc (cz + d) 2 0 Hence, w is conformal at every point z d/c. 2

3 Having in mind this observation and the previous deliberations, we can summarize Theorem 5.1. Let f be a Möbius transformation. Then f can be expressed as a composition of magnifications, rotations, translations and inversions. f maps the extended complex plane onto itself. f maps the class of circled and lines to itself. f is conformal at every point except its pole The group of Möbius transformations. Let w = f(z) = az + b cz + d (7) be a möbius transformation. The inverse function z = f 1 (w) (that is: f f 1 I, I the identity) can be computed directly: f 1 (w) = dw b cw + a. We see that the inverse is again a Möbius transformation. Furthermore, we easily check that the composition of two transformations f 1 f 2 (z) (e.g. if then f i (z) = a iz + b i c i z + d i, i = 1, 2 f 1 f 2 (z) := f 1 (f 2 (z)) = a 1f 2 (z) + b 1 c 1 f 2 (z) + d 2 ) is again a transformation of Möbius. On the other hand, f I(z) = I f(z) = f(z). So, the set of all Möbius transformations is a group with respect to the composition. Now we shall prove Theorem 5.2. A Möbius transformation is uniquely determined by three points z i, i = 1, 2, 3, z i z j, i, j = 1, 2, 3. Proof: We first introduce the term of a double point. We say that z 0 is double point of f(z), if f(z) = z. (8) 3

4 It It is obvious that a Möbius transformation has not more that two double points unless it coincides with the identity. Indeed, if (8) is fulfilled for three distinct points, then the quadratic equation az + b = cz 2 + dz will have three distinct zeros, which implies a = d, b = c = 0. Notice that a linear transformation has only one double point. Let now z i, i = 1, 2, 3 and w i, i = 1, 2, 3 be given, z i z j, w i w j, i.j. = 1, 2, 3. We are looking for a transformation w = f(z) such that f(z i ) = w i. (9) Consider the cross-ratio (z, z 1, z 2, z 3 ) of the points z, z i, i = 1, 2, 3, that is T (z) = (z, z 1, z 2, z 3 ) := (z z 1)(z 2 z 3 ) (z z 3 )(z 2 z 1 ). The function T (z), z C, z i fixed maps C in a one-to-one way onto itself 1. Notice that the order in which the points are listed is crucial in this notation. Then the desired transformation (9) is given by the composition (z, z 1, z 2, z 3 ) = (w, w 1, w 2, w 3 ). It remains only to equate w. The next step is to show that w = f(z) is the only transformation with property (9). Indeed, suppose to the contrary that there is another Möbius transformation v = g(z), f g which satisfies (9). Then the Möbius transformation f g 1 has three distinct double points which is impossible. Q.E.D. Example 5.1: Find a Möbius transformation T (z) that maps the real line R onto the unit circle C 0 (1). Solution: We select three arbitrary real points, say 1, 0, 1 and three arbitrary points on C 0 (1), say i, 1, i. The transformation w = T (z), determined by the cross products, namely unity (z, 1, 0, 1) = (w, i, 1, i) 1 if some number z i equals infinity, then the factors containing it are replaced by the 4

5 maps the real line R onto the circle C passing through the points i, 1, i. As we know from the elementary geometry, C C 0 (1). ℵ Example 5.2: Let w(z) := z + 1 z 1. Find the image of R, I and C 0 (1) under the mapping w = T (z). Solution: We first observe that Further, we see that R R. T ( 1) = 0, T (1) = T (0) = 1, T ( ) = 1, T (i) = i. The image of I is a line or a circle passing through the points 1, 1, i, that is the unit circle. The image of C 0 (1) is a line or a circle that passes through 0, and is orthogonal to the images of R and of I, in our case orthogonal to R and to C 0 (1). This turns to be the imaginary axis The left-hand-rule. Consider the unit circle C 0 (1). The points 1, i, 1 determine the direction 1 i 1 1 in traversing C 0 (1). The interior of the circle, the unit disk D 0 (1) lies to the left of this orientation. We use to say that the disk is the left region with respect to the orientation 1 i 1. Analogously, the upper half plane is the left region with respect to the direction 1, 0,. Since Möbius transformations transformations are conformal mappings, it can be shown that a Möbius transformation that takes the distinct points z 1, z 2, z 3 to the respective points w 1, w 2, w 3 must map the left region of the circle (or line) oriented by z 1, z 2, z 3 onto the left region of the circle (or line) oriented by w 1, w 2, w 3. Using the conformality, we summarize Theorem 5.3. Let G be a domain in C, G := Γ and assume that G is left oriented with respect to the direction given by the points z 1 z 2 z 3, z i Γ, i = 1, 2, 3 Γ. Let w = T (z) be a Möbius transformationthat maps Γ onto γ and T (z i ) = w i, i = 1, 2, 3. Then T maps G onto that domain which is left oriented to γ with respect to the direction w 1 w 2 w 3. 5

6 Example 5.3: Let w = T (z) be the Möbius transformationfrom Example 2. Find the images of the upper half plane, of the left half plane and of the unit disk. Solution: As we have seen, T : R R, I C 0 (1), C 0 (1) I. The upper plane the left domain with respect to the direction 1 0, hence the range domain will be left oriented with respect to 0, 1, 1 (the images of 1, 0, ), e.g., the half plane below the real axis. Similarly, we find that the left half plane is mapped in the unit disk, whereas the unit disk - in the left half plane The symmetry and the Möbius transformation. Definition: Given the circle C a (r) of radius r and centered at z = a, we say that the points z, z C a (r) are symmetric with respect to C a (r) if z = { R 2 + a, z a, z ā, z = a. One can easily prove that in this definition each straight line or circle passing through z and z intersects tc a (r) orthogonally. In the case of a line we have the usual symmetry with respect to it. Theorem 5.3. Given a circle C and a Möbius transformationw = T (z), assume that z, z are symmetric with respect to C. Then their images T (z), T (z ) are again symmetric with respect to the image T (C) of C. Example 5.4: Find all Möbius transformationtransformations that map the disk C 0 (r) onto itself. Solution; We fix in an arbitrary way a point a D 0 (r). Its symmetric point with respect to the unit circle is Any transformation of the form a = r2 ā. (10) S(z) := K z a z r 2 a 6 (11)

7 with K being some constant maps the points a and a at 0 and infinity, respectively. Because of the symmetry, the image of C 0 (r) will be a circle C centered at the zero. Now we are going to choose the constant K in such a way that C coincides with C 0 (r). Take a point z 0 := re iφ and calculate S(z 0 ). We have, by (10), S(z 0 ) = K reiφ a a re iφ r, 2 or Putting we arrive at for some real Θ. Exercises: 1. Let S(z 0 ) = K r S(z 0 ) = r, re iφ a a re iφ. T (z) = r 2 e iθ z a z r 2 a w = 1/z. Find the image of the lines y = kx, y = ax + b, and of the circle x 2 + y 2 = ax, x 2 + y Let Find the image of {x, y 0}. w = z i z + i. 3. Let w = z z 1. Find the image of the angle 0 φ π 4 4. Find all Möbius transformations that maps the upper half plane onto itself. 5. Given T (z) := z 1 z+1, find the image of the domain G := {z, z < 1} {z, z 1 < 1}. 7

Selected practice exam solutions (part 5, item 2) (MAT 360)

Selected practice exam solutions (part 5, item 2) (MAT 360) Selected practice exam solutions (part 5, item ) (MAT 360) Harder 8,91,9,94(smaller should be replaced by greater )95,103,109,140,160,(178,179,180,181 this is really one problem),188,193,194,195 8. On

More information

MATH 304 Linear Algebra Lecture 24: Scalar product.

MATH 304 Linear Algebra Lecture 24: Scalar product. MATH 304 Linear Algebra Lecture 24: Scalar product. Vectors: geometric approach B A B A A vector is represented by a directed segment. Directed segment is drawn as an arrow. Different arrows represent

More information

Lecture 2: Homogeneous Coordinates, Lines and Conics

Lecture 2: Homogeneous Coordinates, Lines and Conics Lecture 2: Homogeneous Coordinates, Lines and Conics 1 Homogeneous Coordinates In Lecture 1 we derived the camera equations λx = P X, (1) where x = (x 1, x 2, 1), X = (X 1, X 2, X 3, 1) and P is a 3 4

More information

7. Cauchy s integral theorem and Cauchy s integral formula

7. Cauchy s integral theorem and Cauchy s integral formula 7. Cauchy s integral theorem and Cauchy s integral formula 7.. Independence of the path of integration Theorem 6.3. can be rewritten in the following form: Theorem 7. : Let D be a domain in C and suppose

More information

1 Review of complex numbers

1 Review of complex numbers 1 Review of complex numbers 1.1 Complex numbers: algebra The set C of complex numbers is formed by adding a square root i of 1 to the set of real numbers: i = 1. Every complex number can be written uniquely

More information

Solutions to Practice Problems

Solutions to Practice Problems Higher Geometry Final Exam Tues Dec 11, 5-7:30 pm Practice Problems (1) Know the following definitions, statements of theorems, properties from the notes: congruent, triangle, quadrilateral, isosceles

More information

Section 9.5: Equations of Lines and Planes

Section 9.5: Equations of Lines and Planes Lines in 3D Space Section 9.5: Equations of Lines and Planes Practice HW from Stewart Textbook (not to hand in) p. 673 # 3-5 odd, 2-37 odd, 4, 47 Consider the line L through the point P = ( x, y, ) that

More information

Casey s Theorem and its Applications

Casey s Theorem and its Applications Casey s Theorem and its Applications Luis González Maracaibo. Venezuela July 011 Abstract. We present a proof of the generalized Ptolemy s theorem, also known as Casey s theorem and its applications in

More information

POWER SETS AND RELATIONS

POWER SETS AND RELATIONS POWER SETS AND RELATIONS L. MARIZZA A. BAILEY 1. The Power Set Now that we have defined sets as best we can, we can consider a sets of sets. If we were to assume nothing, except the existence of the empty

More information

Lecture 8 : Coordinate Geometry. The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 20

Lecture 8 : Coordinate Geometry. The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 20 Lecture 8 : Coordinate Geometry The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 0 distance on the axis and give each point an identity on the corresponding

More information

DEFINITION 5.1.1 A complex number is a matrix of the form. x y. , y x

DEFINITION 5.1.1 A complex number is a matrix of the form. x y. , y x Chapter 5 COMPLEX NUMBERS 5.1 Constructing the complex numbers One way of introducing the field C of complex numbers is via the arithmetic of matrices. DEFINITION 5.1.1 A complex number is a matrix of

More information

Double Tangent Circles and Focal Properties of Sphero-Conics

Double Tangent Circles and Focal Properties of Sphero-Conics Double Tangent Circles and Focal Properties of Sphero-Conics Hans-Peter Schröcker University of Innsbruck Unit Geometry and CAD April 10, 008 Abstract We give two proofs for the characterization of a sphero-conic

More information

11.1. Objectives. Component Form of a Vector. Component Form of a Vector. Component Form of a Vector. Vectors and the Geometry of Space

11.1. Objectives. Component Form of a Vector. Component Form of a Vector. Component Form of a Vector. Vectors and the Geometry of Space 11 Vectors and the Geometry of Space 11.1 Vectors in the Plane Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. 2 Objectives! Write the component form of

More information

n th roots of complex numbers

n th roots of complex numbers n th roots of complex numbers Nathan Pflueger 1 October 014 This note describes how to solve equations of the form z n = c, where c is a complex number. These problems serve to illustrate the use of polar

More information

WHICH LINEAR-FRACTIONAL TRANSFORMATIONS INDUCE ROTATIONS OF THE SPHERE?

WHICH LINEAR-FRACTIONAL TRANSFORMATIONS INDUCE ROTATIONS OF THE SPHERE? WHICH LINEAR-FRACTIONAL TRANSFORMATIONS INDUCE ROTATIONS OF THE SPHERE? JOEL H. SHAPIRO Abstract. These notes supplement the discussion of linear fractional mappings presented in a beginning graduate course

More information

PYTHAGOREAN TRIPLES PETE L. CLARK

PYTHAGOREAN TRIPLES PETE L. CLARK PYTHAGOREAN TRIPLES PETE L. CLARK 1. Parameterization of Pythagorean Triples 1.1. Introduction to Pythagorean triples. By a Pythagorean triple we mean an ordered triple (x, y, z) Z 3 such that x + y =

More information

2 Complex Functions and the Cauchy-Riemann Equations

2 Complex Functions and the Cauchy-Riemann Equations 2 Complex Functions and the Cauchy-Riemann Equations 2.1 Complex functions In one-variable calculus, we study functions f(x) of a real variable x. Likewise, in complex analysis, we study functions f(z)

More information

1. LINEAR EQUATIONS. A linear equation in n unknowns x 1, x 2,, x n is an equation of the form

1. LINEAR EQUATIONS. A linear equation in n unknowns x 1, x 2,, x n is an equation of the form 1. LINEAR EQUATIONS A linear equation in n unknowns x 1, x 2,, x n is an equation of the form a 1 x 1 + a 2 x 2 + + a n x n = b, where a 1, a 2,..., a n, b are given real numbers. For example, with x and

More information

Lecture 17: Conformal Invariance

Lecture 17: Conformal Invariance Lecture 17: Conformal Invariance Scribe: Yee Lok Wong Department of Mathematics, MIT November 7, 006 1 Eventual Hitting Probability In previous lectures, we studied the following PDE for ρ(x, t x 0 ) that

More information

9 Multiplication of Vectors: The Scalar or Dot Product

9 Multiplication of Vectors: The Scalar or Dot Product Arkansas Tech University MATH 934: Calculus III Dr. Marcel B Finan 9 Multiplication of Vectors: The Scalar or Dot Product Up to this point we have defined what vectors are and discussed basic notation

More information

Change of Continuous Random Variable

Change of Continuous Random Variable Change of Continuous Random Variable All you are responsible for from this lecture is how to implement the Engineer s Way (see page 4) to compute how the probability density function changes when we make

More information

The Stern-Gerlach Experiment

The Stern-Gerlach Experiment Chapter The Stern-Gerlach Experiment Let us now talk about a particular property of an atom, called its magnetic dipole moment. It is simplest to first recall what an electric dipole moment is. Consider

More information

Diagonal, Symmetric and Triangular Matrices

Diagonal, Symmetric and Triangular Matrices Contents 1 Diagonal, Symmetric Triangular Matrices 2 Diagonal Matrices 2.1 Products, Powers Inverses of Diagonal Matrices 2.1.1 Theorem (Powers of Matrices) 2.2 Multiplying Matrices on the Left Right by

More information

Applications of Fermat s Little Theorem and Congruences

Applications of Fermat s Little Theorem and Congruences Applications of Fermat s Little Theorem and Congruences Definition: Let m be a positive integer. Then integers a and b are congruent modulo m, denoted by a b mod m, if m (a b). Example: 3 1 mod 2, 6 4

More information

Chapter 17. Orthogonal Matrices and Symmetries of Space

Chapter 17. Orthogonal Matrices and Symmetries of Space Chapter 17. Orthogonal Matrices and Symmetries of Space Take a random matrix, say 1 3 A = 4 5 6, 7 8 9 and compare the lengths of e 1 and Ae 1. The vector e 1 has length 1, while Ae 1 = (1, 4, 7) has length

More information

Algebra 2 Chapter 1 Vocabulary. identity - A statement that equates two equivalent expressions.

Algebra 2 Chapter 1 Vocabulary. identity - A statement that equates two equivalent expressions. Chapter 1 Vocabulary identity - A statement that equates two equivalent expressions. verbal model- A word equation that represents a real-life problem. algebraic expression - An expression with variables.

More information

Residues and Contour Integration Problems

Residues and Contour Integration Problems Residues and ontour Integration Problems lassify the singularity of f(z) at the indicated point.. f(z) = cot(z) at z =. Ans. Simple pole. Solution. The test for a simple pole at z = is that lim z zcot(z)

More information

5.3 Improper Integrals Involving Rational and Exponential Functions

5.3 Improper Integrals Involving Rational and Exponential Functions Section 5.3 Improper Integrals Involving Rational and Exponential Functions 99.. 3. 4. dθ +a cos θ =, < a

More information

THREE DIMENSIONAL GEOMETRY

THREE DIMENSIONAL GEOMETRY Chapter 8 THREE DIMENSIONAL GEOMETRY 8.1 Introduction In this chapter we present a vector algebra approach to three dimensional geometry. The aim is to present standard properties of lines and planes,

More information

2D Geometric Transformations. COMP 770 Fall 2011

2D Geometric Transformations. COMP 770 Fall 2011 2D Geometric Transformations COMP 770 Fall 2011 1 A little quick math background Notation for sets, functions, mappings Linear transformations Matrices Matrix-vector multiplication Matrix-matrix multiplication

More information

88 CHAPTER 2. VECTOR FUNCTIONS. . First, we need to compute T (s). a By definition, r (s) T (s) = 1 a sin s a. sin s a, cos s a

88 CHAPTER 2. VECTOR FUNCTIONS. . First, we need to compute T (s). a By definition, r (s) T (s) = 1 a sin s a. sin s a, cos s a 88 CHAPTER. VECTOR FUNCTIONS.4 Curvature.4.1 Definitions and Examples The notion of curvature measures how sharply a curve bends. We would expect the curvature to be 0 for a straight line, to be very small

More information

What is inversive geometry?

What is inversive geometry? What is inversive geometry? Andrew Krieger July 18, 2013 Throughout, Greek letters (,,...) denote geometric objects like circles or lines; small Roman letters (a, b,... ) denote distances, and large Roman

More information

Geometric Transformations

Geometric Transformations Geometric Transformations Definitions Def: f is a mapping (function) of a set A into a set B if for every element a of A there exists a unique element b of B that is paired with a; this pairing is denoted

More information

Section 1.1. Introduction to R n

Section 1.1. Introduction to R n The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to

More information

Geometry of Vectors. 1 Cartesian Coordinates. Carlo Tomasi

Geometry of Vectors. 1 Cartesian Coordinates. Carlo Tomasi Geometry of Vectors Carlo Tomasi This note explores the geometric meaning of norm, inner product, orthogonality, and projection for vectors. For vectors in three-dimensional space, we also examine the

More information

3 Contour integrals and Cauchy s Theorem

3 Contour integrals and Cauchy s Theorem 3 ontour integrals and auchy s Theorem 3. Line integrals of complex functions Our goal here will be to discuss integration of complex functions = u + iv, with particular regard to analytic functions. Of

More information

a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.

a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2. Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given

More information

Euclidean Geometry. We start with the idea of an axiomatic system. An axiomatic system has four parts:

Euclidean Geometry. We start with the idea of an axiomatic system. An axiomatic system has four parts: Euclidean Geometry Students are often so challenged by the details of Euclidean geometry that they miss the rich structure of the subject. We give an overview of a piece of this structure below. We start

More information

Numerical Analysis Lecture Notes

Numerical Analysis Lecture Notes Numerical Analysis Lecture Notes Peter J. Olver 5. Inner Products and Norms The norm of a vector is a measure of its size. Besides the familiar Euclidean norm based on the dot product, there are a number

More information

v w is orthogonal to both v and w. the three vectors v, w and v w form a right-handed set of vectors.

v w is orthogonal to both v and w. the three vectors v, w and v w form a right-handed set of vectors. 3. Cross product Definition 3.1. Let v and w be two vectors in R 3. The cross product of v and w, denoted v w, is the vector defined as follows: the length of v w is the area of the parallelogram with

More information

Notes on Determinant

Notes on Determinant ENGG2012B Advanced Engineering Mathematics Notes on Determinant Lecturer: Kenneth Shum Lecture 9-18/02/2013 The determinant of a system of linear equations determines whether the solution is unique, without

More information

Linear Algebra Notes for Marsden and Tromba Vector Calculus

Linear Algebra Notes for Marsden and Tromba Vector Calculus Linear Algebra Notes for Marsden and Tromba Vector Calculus n-dimensional Euclidean Space and Matrices Definition of n space As was learned in Math b, a point in Euclidean three space can be thought of

More information

6. Define log(z) so that π < I log(z) π. Discuss the identities e log(z) = z and log(e w ) = w.

6. Define log(z) so that π < I log(z) π. Discuss the identities e log(z) = z and log(e w ) = w. hapter omplex integration. omplex number quiz. Simplify 3+4i. 2. Simplify 3+4i. 3. Find the cube roots of. 4. Here are some identities for complex conjugate. Which ones need correction? z + w = z + w,

More information

Examples of Functions

Examples of Functions Chapter 3 Examples of Functions Obvious is the most dangerous word in mathematics. E. T. Bell 3.1 Möbius Transformations The first class of functions that we will discuss in some detail are built from

More information

Chapter 4.1 Parallel Lines and Planes

Chapter 4.1 Parallel Lines and Planes Chapter 4.1 Parallel Lines and Planes Expand on our definition of parallel lines Introduce the idea of parallel planes. What do we recall about parallel lines? In geometry, we have to be concerned about

More information

NOTES ON LINEAR TRANSFORMATIONS

NOTES ON LINEAR TRANSFORMATIONS NOTES ON LINEAR TRANSFORMATIONS Definition 1. Let V and W be vector spaces. A function T : V W is a linear transformation from V to W if the following two properties hold. i T v + v = T v + T v for all

More information

LINEAR ALGEBRA W W L CHEN

LINEAR ALGEBRA W W L CHEN LINEAR ALGEBRA W W L CHEN c W W L Chen, 1997, 2008 This chapter is available free to all individuals, on understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied,

More information

Trigonometric Functions and Equations

Trigonometric Functions and Equations Contents Trigonometric Functions and Equations Lesson 1 Reasoning with Trigonometric Functions Investigations 1 Proving Trigonometric Identities... 271 2 Sum and Difference Identities... 276 3 Extending

More information

12.5 Equations of Lines and Planes

12.5 Equations of Lines and Planes Instructor: Longfei Li Math 43 Lecture Notes.5 Equations of Lines and Planes What do we need to determine a line? D: a point on the line: P 0 (x 0, y 0 ) direction (slope): k 3D: a point on the line: P

More information

Proofs of the inscribed angle theorem

Proofs of the inscribed angle theorem I2GEO Proofs of the inscribed angle theorem University of South Bohemia Faculty of Education Jeronýmova 10 371 15 České Budějovice Czech Republic E-mail adress: leischne@pf.jcu.cz Abstract The inscribed

More information

5.4 The Quadratic Formula

5.4 The Quadratic Formula Section 5.4 The Quadratic Formula 481 5.4 The Quadratic Formula Consider the general quadratic function f(x) = ax + bx + c. In the previous section, we learned that we can find the zeros of this function

More information

We call this set an n-dimensional parallelogram (with one vertex 0). We also refer to the vectors x 1,..., x n as the edges of P.

We call this set an n-dimensional parallelogram (with one vertex 0). We also refer to the vectors x 1,..., x n as the edges of P. Volumes of parallelograms 1 Chapter 8 Volumes of parallelograms In the present short chapter we are going to discuss the elementary geometrical objects which we call parallelograms. These are going to

More information

1 Solution of Homework

1 Solution of Homework Math 3181 Dr. Franz Rothe February 4, 2011 Name: 1 Solution of Homework 10 Problem 1.1 (Common tangents of two circles). How many common tangents do two circles have. Informally draw all different cases,

More information

Orthogonal Projections

Orthogonal Projections Orthogonal Projections and Reflections (with exercises) by D. Klain Version.. Corrections and comments are welcome! Orthogonal Projections Let X,..., X k be a family of linearly independent (column) vectors

More information

3. Let A and B be two n n orthogonal matrices. Then prove that AB and BA are both orthogonal matrices. Prove a similar result for unitary matrices.

3. Let A and B be two n n orthogonal matrices. Then prove that AB and BA are both orthogonal matrices. Prove a similar result for unitary matrices. Exercise 1 1. Let A be an n n orthogonal matrix. Then prove that (a) the rows of A form an orthonormal basis of R n. (b) the columns of A form an orthonormal basis of R n. (c) for any two vectors x,y R

More information

Introduction. Modern Applications Computer Graphics Bio-medical Applications Modeling Cryptology and Coding Theory

Introduction. Modern Applications Computer Graphics Bio-medical Applications Modeling Cryptology and Coding Theory Higher Geometry Introduction Brief Historical Sketch Greeks (Thales, Euclid, Archimedes) Parallel Postulate Non-Euclidean Geometries Projective Geometry Revolution Axiomatics revisited Modern Geometries

More information

Adding vectors We can do arithmetic with vectors. We ll start with vector addition and related operations. Suppose you have two vectors

Adding vectors We can do arithmetic with vectors. We ll start with vector addition and related operations. Suppose you have two vectors 1 Chapter 13. VECTORS IN THREE DIMENSIONAL SPACE Let s begin with some names and notation for things: R is the set (collection) of real numbers. We write x R to mean that x is a real number. A real number

More information

Equations Involving Lines and Planes Standard equations for lines in space

Equations Involving Lines and Planes Standard equations for lines in space Equations Involving Lines and Planes In this section we will collect various important formulas regarding equations of lines and planes in three dimensional space Reminder regarding notation: any quantity

More information

Figure 1.1 Vector A and Vector F

Figure 1.1 Vector A and Vector F CHAPTER I VECTOR QUANTITIES Quantities are anything which can be measured, and stated with number. Quantities in physics are divided into two types; scalar and vector quantities. Scalar quantities have

More information

* Biot Savart s Law- Statement, Proof Applications of Biot Savart s Law * Magnetic Field Intensity H * Divergence of B * Curl of B. PPT No.

* Biot Savart s Law- Statement, Proof Applications of Biot Savart s Law * Magnetic Field Intensity H * Divergence of B * Curl of B. PPT No. * Biot Savart s Law- Statement, Proof Applications of Biot Savart s Law * Magnetic Field Intensity H * Divergence of B * Curl of B PPT No. 17 Biot Savart s Law A straight infinitely long wire is carrying

More information

Lecture L22-2D Rigid Body Dynamics: Work and Energy

Lecture L22-2D Rigid Body Dynamics: Work and Energy J. Peraire, S. Widnall 6.07 Dynamics Fall 008 Version.0 Lecture L - D Rigid Body Dynamics: Work and Energy In this lecture, we will revisit the principle of work and energy introduced in lecture L-3 for

More information

alternate interior angles

alternate interior angles alternate interior angles two non-adjacent angles that lie on the opposite sides of a transversal between two lines that the transversal intersects (a description of the location of the angles); alternate

More information

Lesson 19: Equations for Tangent Lines to Circles

Lesson 19: Equations for Tangent Lines to Circles Student Outcomes Given a circle, students find the equations of two lines tangent to the circle with specified slopes. Given a circle and a point outside the circle, students find the equation of the line

More information

Vector Spaces II: Finite Dimensional Linear Algebra 1

Vector Spaces II: Finite Dimensional Linear Algebra 1 John Nachbar September 2, 2014 Vector Spaces II: Finite Dimensional Linear Algebra 1 1 Definitions and Basic Theorems. For basic properties and notation for R N, see the notes Vector Spaces I. Definition

More information

Coefficient of Potential and Capacitance

Coefficient of Potential and Capacitance Coefficient of Potential and Capacitance Lecture 12: Electromagnetic Theory Professor D. K. Ghosh, Physics Department, I.I.T., Bombay We know that inside a conductor there is no electric field and that

More information

5.3 The Cross Product in R 3

5.3 The Cross Product in R 3 53 The Cross Product in R 3 Definition 531 Let u = [u 1, u 2, u 3 ] and v = [v 1, v 2, v 3 ] Then the vector given by [u 2 v 3 u 3 v 2, u 3 v 1 u 1 v 3, u 1 v 2 u 2 v 1 ] is called the cross product (or

More information

Foundations of Geometry 1: Points, Lines, Segments, Angles

Foundations of Geometry 1: Points, Lines, Segments, Angles Chapter 3 Foundations of Geometry 1: Points, Lines, Segments, Angles 3.1 An Introduction to Proof Syllogism: The abstract form is: 1. All A is B. 2. X is A 3. X is B Example: Let s think about an example.

More information

S on n elements. A good way to think about permutations is the following. Consider the A = 1,2,3, 4 whose elements we permute with the P =

S on n elements. A good way to think about permutations is the following. Consider the A = 1,2,3, 4 whose elements we permute with the P = Section 6. 1 Section 6. Groups of Permutations: : The Symmetric Group Purpose of Section: To introduce the idea of a permutation and show how the set of all permutations of a set of n elements, equipped

More information

Review of Fundamental Mathematics

Review of Fundamental Mathematics Review of Fundamental Mathematics As explained in the Preface and in Chapter 1 of your textbook, managerial economics applies microeconomic theory to business decision making. The decision-making tools

More information

Math 241, Exam 1 Information.

Math 241, Exam 1 Information. Math 241, Exam 1 Information. 9/24/12, LC 310, 11:15-12:05. Exam 1 will be based on: Sections 12.1-12.5, 14.1-14.3. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/241fa12/241.html)

More information

4. MATRICES Matrices

4. MATRICES Matrices 4. MATRICES 170 4. Matrices 4.1. Definitions. Definition 4.1.1. A matrix is a rectangular array of numbers. A matrix with m rows and n columns is said to have dimension m n and may be represented as follows:

More information

39 Symmetry of Plane Figures

39 Symmetry of Plane Figures 39 Symmetry of Plane Figures In this section, we are interested in the symmetric properties of plane figures. By a symmetry of a plane figure we mean a motion of the plane that moves the figure so that

More information

Chapter 8 Graphs and Functions:

Chapter 8 Graphs and Functions: Chapter 8 Graphs and Functions: Cartesian axes, coordinates and points 8.1 Pictorially we plot points and graphs in a plane (flat space) using a set of Cartesian axes traditionally called the x and y axes

More information

PROJECTIVE GEOMETRY. b3 course 2003. Nigel Hitchin

PROJECTIVE GEOMETRY. b3 course 2003. Nigel Hitchin PROJECTIVE GEOMETRY b3 course 2003 Nigel Hitchin hitchin@maths.ox.ac.uk 1 1 Introduction This is a course on projective geometry. Probably your idea of geometry in the past has been based on triangles

More information

ELEMENTS OF VECTOR ALGEBRA

ELEMENTS OF VECTOR ALGEBRA ELEMENTS OF VECTOR ALGEBRA A.1. VECTORS AND SCALAR QUANTITIES We have now proposed sets of basic dimensions and secondary dimensions to describe certain aspects of nature, but more than just dimensions

More information

Vector Notation: AB represents the vector from point A to point B on a graph. The vector can be computed by B A.

Vector Notation: AB represents the vector from point A to point B on a graph. The vector can be computed by B A. 1 Linear Transformations Prepared by: Robin Michelle King A transformation of an object is a change in position or dimension (or both) of the object. The resulting object after the transformation is called

More information

Pythagorean Triples. Chapter 2. a 2 + b 2 = c 2

Pythagorean Triples. Chapter 2. a 2 + b 2 = c 2 Chapter Pythagorean Triples The Pythagorean Theorem, that beloved formula of all high school geometry students, says that the sum of the squares of the sides of a right triangle equals the square of the

More information

Solutions to Homework 10

Solutions to Homework 10 Solutions to Homework 1 Section 7., exercise # 1 (b,d): (b) Compute the value of R f dv, where f(x, y) = y/x and R = [1, 3] [, 4]. Solution: Since f is continuous over R, f is integrable over R. Let x

More information

Elementary triangle geometry

Elementary triangle geometry Elementary triangle geometry Dennis Westra March 26, 2010 bstract In this short note we discuss some fundamental properties of triangles up to the construction of the Euler line. ontents ngle bisectors

More information

Understanding Basic Calculus

Understanding Basic Calculus Understanding Basic Calculus S.K. Chung Dedicated to all the people who have helped me in my life. i Preface This book is a revised and expanded version of the lecture notes for Basic Calculus and other

More information

x1 x 2 x 3 y 1 y 2 y 3 x 1 y 2 x 2 y 1 0.

x1 x 2 x 3 y 1 y 2 y 3 x 1 y 2 x 2 y 1 0. Cross product 1 Chapter 7 Cross product We are getting ready to study integration in several variables. Until now we have been doing only differential calculus. One outcome of this study will be our ability

More information

1 of 79 Erik Eberhardt UBC Geological Engineering EOSC 433

1 of 79 Erik Eberhardt UBC Geological Engineering EOSC 433 Stress & Strain: A review xx yz zz zx zy xy xz yx yy xx yy zz 1 of 79 Erik Eberhardt UBC Geological Engineering EOSC 433 Disclaimer before beginning your problem assignment: Pick up and compare any set

More information

Recall that the gradient of a differentiable scalar field ϕ on an open set D in R n is given by the formula:

Recall that the gradient of a differentiable scalar field ϕ on an open set D in R n is given by the formula: Chapter 7 Div, grad, and curl 7.1 The operator and the gradient: Recall that the gradient of a differentiable scalar field ϕ on an open set D in R n is given by the formula: ( ϕ ϕ =, ϕ,..., ϕ. (7.1 x 1

More information

Notes on Orthogonal and Symmetric Matrices MENU, Winter 2013

Notes on Orthogonal and Symmetric Matrices MENU, Winter 2013 Notes on Orthogonal and Symmetric Matrices MENU, Winter 201 These notes summarize the main properties and uses of orthogonal and symmetric matrices. We covered quite a bit of material regarding these topics,

More information

Practice with Proofs

Practice with Proofs Practice with Proofs October 6, 2014 Recall the following Definition 0.1. A function f is increasing if for every x, y in the domain of f, x < y = f(x) < f(y) 1. Prove that h(x) = x 3 is increasing, using

More information

Dot product and vector projections (Sect. 12.3) There are two main ways to introduce the dot product

Dot product and vector projections (Sect. 12.3) There are two main ways to introduce the dot product Dot product and vector projections (Sect. 12.3) Two definitions for the dot product. Geometric definition of dot product. Orthogonal vectors. Dot product and orthogonal projections. Properties of the dot

More information

Diagonalisation. Chapter 3. Introduction. Eigenvalues and eigenvectors. Reading. Definitions

Diagonalisation. Chapter 3. Introduction. Eigenvalues and eigenvectors. Reading. Definitions Chapter 3 Diagonalisation Eigenvalues and eigenvectors, diagonalisation of a matrix, orthogonal diagonalisation fo symmetric matrices Reading As in the previous chapter, there is no specific essential

More information

Vectors, Gradient, Divergence and Curl.

Vectors, Gradient, Divergence and Curl. Vectors, Gradient, Divergence and Curl. 1 Introduction A vector is determined by its length and direction. They are usually denoted with letters with arrows on the top a or in bold letter a. We will use

More information

INCIDENCE-BETWEENNESS GEOMETRY

INCIDENCE-BETWEENNESS GEOMETRY INCIDENCE-BETWEENNESS GEOMETRY MATH 410, CSUSM. SPRING 2008. PROFESSOR AITKEN This document covers the geometry that can be developed with just the axioms related to incidence and betweenness. The full

More information

TOPIC 4: DERIVATIVES

TOPIC 4: DERIVATIVES TOPIC 4: DERIVATIVES 1. The derivative of a function. Differentiation rules 1.1. The slope of a curve. The slope of a curve at a point P is a measure of the steepness of the curve. If Q is a point on the

More information

Matrix Representations of Linear Transformations and Changes of Coordinates

Matrix Representations of Linear Transformations and Changes of Coordinates Matrix Representations of Linear Transformations and Changes of Coordinates 01 Subspaces and Bases 011 Definitions A subspace V of R n is a subset of R n that contains the zero element and is closed under

More information

USING VECTORS TO MEASURE ANGLES BETWEEN LINES IN SPACE

USING VECTORS TO MEASURE ANGLES BETWEEN LINES IN SPACE USING VECTORS TO MEASURE ANGLES BETWEEN LINES IN SPACE Consider a straight line in Cartesian D space [x,y,z]. Let two points on the line e [x,y,z ] and [x,y,z ]. The slopes of this line are constants and

More information

MAT 200, Midterm Exam Solution. a. (5 points) Compute the determinant of the matrix A =

MAT 200, Midterm Exam Solution. a. (5 points) Compute the determinant of the matrix A = MAT 200, Midterm Exam Solution. (0 points total) a. (5 points) Compute the determinant of the matrix 2 2 0 A = 0 3 0 3 0 Answer: det A = 3. The most efficient way is to develop the determinant along the

More information

The Math Circle, Spring 2004

The Math Circle, Spring 2004 The Math Circle, Spring 2004 (Talks by Gordon Ritter) What is Non-Euclidean Geometry? Most geometries on the plane R 2 are non-euclidean. Let s denote arc length. Then Euclidean geometry arises from the

More information

FURTHER VECTORS (MEI)

FURTHER VECTORS (MEI) Mathematics Revision Guides Further Vectors (MEI) (column notation) Page of MK HOME TUITION Mathematics Revision Guides Level: AS / A Level - MEI OCR MEI: C FURTHER VECTORS (MEI) Version : Date: -9-7 Mathematics

More information

Calculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum

Calculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum Calculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum UNIT I: The Hyperbolic Functions basic calculus concepts, including techniques for curve sketching, exponential and logarithmic

More information

3. Reaction Diffusion Equations Consider the following ODE model for population growth

3. Reaction Diffusion Equations Consider the following ODE model for population growth 3. Reaction Diffusion Equations Consider the following ODE model for population growth u t a u t u t, u 0 u 0 where u t denotes the population size at time t, and a u plays the role of the population dependent

More information

Roots and Coefficients of a Quadratic Equation Summary

Roots and Coefficients of a Quadratic Equation Summary Roots and Coefficients of a Quadratic Equation Summary For a quadratic equation with roots α and β: Sum of roots = α + β = and Product of roots = αβ = Symmetrical functions of α and β include: x = and

More information

m i: is the mass of each particle

m i: is the mass of each particle Center of Mass (CM): The center of mass is a point which locates the resultant mass of a system of particles or body. It can be within the object (like a human standing straight) or outside the object

More information

Chapter 5 Polar Coordinates; Vectors 5.1 Polar coordinates 1. Pole and polar axis

Chapter 5 Polar Coordinates; Vectors 5.1 Polar coordinates 1. Pole and polar axis Chapter 5 Polar Coordinates; Vectors 5.1 Polar coordinates 1. Pole and polar axis 2. Polar coordinates A point P in a polar coordinate system is represented by an ordered pair of numbers (r, θ). If r >

More information